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Theorem funssxp 5571
Description: Two ways of specifying a partial function from  A to  B. (Contributed by NM, 13-Nov-2007.)
Assertion
Ref Expression
funssxp  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  <->  ( F : dom  F --> B  /\  dom  F  C_  A )
)

Proof of Theorem funssxp
StepHypRef Expression
1 funfn 5449 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
21biimpi 187 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
3 rnss 5065 . . . . . 6  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  ran  ( A  X.  B ) )
4 rnxpss 5268 . . . . . 6  |-  ran  ( A  X.  B )  C_  B
53, 4syl6ss 3328 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  ran  F 
C_  B )
62, 5anim12i 550 . . . 4  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F  Fn  dom  F  /\  ran  F  C_  B )
)
7 df-f 5425 . . . 4  |-  ( F : dom  F --> B  <->  ( F  Fn  dom  F  /\  ran  F 
C_  B ) )
86, 7sylibr 204 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  F : dom  F --> B )
9 dmss 5036 . . . . 5  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  dom  ( A  X.  B ) )
10 dmxpss 5267 . . . . 5  |-  dom  ( A  X.  B )  C_  A
119, 10syl6ss 3328 . . . 4  |-  ( F 
C_  ( A  X.  B )  ->  dom  F 
C_  A )
1211adantl 453 . . 3  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  dom  F 
C_  A )
138, 12jca 519 . 2  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  ->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
14 ffun 5560 . . . 4  |-  ( F : dom  F --> B  ->  Fun  F )
1514adantr 452 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  Fun  F )
16 fssxp 5569 . . . 4  |-  ( F : dom  F --> B  ->  F  C_  ( dom  F  X.  B ) )
17 xpss1 4951 . . . 4  |-  ( dom 
F  C_  A  ->  ( dom  F  X.  B
)  C_  ( A  X.  B ) )
1816, 17sylan9ss 3329 . . 3  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  F  C_  ( A  X.  B
) )
1915, 18jca 519 . 2  |-  ( ( F : dom  F --> B  /\  dom  F  C_  A )  ->  ( Fun  F  /\  F  C_  ( A  X.  B
) ) )
2013, 19impbii 181 1  |-  ( ( Fun  F  /\  F  C_  ( A  X.  B
) )  <->  ( F : dom  F --> B  /\  dom  F  C_  A )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    C_ wss 3288    X. cxp 4843   dom cdm 4845   ran crn 4846   Fun wfun 5415    Fn wfn 5416   -->wf 5417
This theorem is referenced by:  elpm2g  7000  volf  19386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-fun 5423  df-fn 5424  df-f 5425
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